Abstract

The two basic optical Fourier transform configurations are examined with respect to component complexity, aberrations, and optical noise. It is shown that the converging-beam illumination setup (CB-FT) is much simpler and works better than the classical parallel beam illumination setup within a restricted range of object size and lens aperture. This range corresponds to many practical cases. Therefore, the CB-FT should be preferred in ordinary cases whereas the classical setup with a special purpose Fourier lens should be used only for a large space–bandwidth product. It is probably never a good solution to use the parallel beam configuration with a general purpose lens as the Fourier lens.

© 1982 Optical Society of America

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References

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  1. H. R. Arsenault, Appl. Opt. 11, 2228 (1972).
    [Crossref] [PubMed]
  2. D. Casasent, T. Luu, Appl. Opt. 17, 2973 (1978).
    [Crossref] [PubMed]
  3. D. Casasent, T. Luu, Appl. Opt. 17, 1701 (1978).
    [Crossref] [PubMed]
  4. P. S. Naidu, D. V. B. Rao, Optik 55, 351 (1979).
  5. M. Stark, Application of Optical Fourier Transform (Academic, New York, 1982), Chap. 1.
  6. According to J. W. Goodman, use of the parallel beam system as an optical Fourier transform was reported first in E. Abbe, Arch. Mikrosk. Anat. Entwicklungsmech. 9, 413 (1873); A. B. Porter, Philos. Mag. (6) 11, 154 (1906). However, the first clear and systematic description of optical Fourier transform is probably P. M Duffieux, “L’intégrale de Fourier et ses applications à l’Optique”, Besançon (1946). All these authors observed the FT in the back focal plane of a lens thus working in parallel beam illumination. The question of the generalization of this setup to any type of source–Fourier plane conjugation is not clear; we found a use for the converging beam setup in A. Marechal and P. Croce, C. R. Acad. Sci. 237, 607 (1953). The question of optical performance was never examined, and the use of optical FT was restricted to very low spatial frequencies and low aperture.
  7. K. Preston, in Optical and electro-optical information processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965), p. 68.
  8. J. W. Goodman, in Optical Information Processing, Y. E. Nesterikhin, G. W. Stroke, W. Kock, Eds. (Plenum, New York, 1976), pp. 85–103.
    [Crossref]
  9. P. Chavel, S. Lowenthal, J. Opt. Soc. Am. 68, 721 (1978).
    [Crossref]
  10. Such a corrected doublet is described in S. Lowenthal and Y. Belvaux, Rev. Opt. Theor. Instrum. 46, 1 (1967).
  11. R. W. Meier, J. Opt. Soc. Am. 55, 987 (1965). We have slightly changed Meier’s Eq. (19) by altering the sign of the wave front error and the definition of astigmatism and field curvature (the sum astigmatism + field curvature remains unchanged, except for the sign).
    [Crossref]
  12. E. B. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
    [Crossref]

1979 (1)

P. S. Naidu, D. V. B. Rao, Optik 55, 351 (1979).

1978 (3)

1972 (1)

1967 (1)

E. B. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
[Crossref]

1965 (1)

Arsenault, H. R.

Casasent, D.

Champagne, E. B.

E. B. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
[Crossref]

Chavel, P.

Goodman, J. W.

J. W. Goodman, in Optical Information Processing, Y. E. Nesterikhin, G. W. Stroke, W. Kock, Eds. (Plenum, New York, 1976), pp. 85–103.
[Crossref]

Lowenthal, S.

Luu, T.

Meier, R. W.

Naidu, P. S.

P. S. Naidu, D. V. B. Rao, Optik 55, 351 (1979).

Preston, K.

K. Preston, in Optical and electro-optical information processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965), p. 68.

Rao, D. V. B.

P. S. Naidu, D. V. B. Rao, Optik 55, 351 (1979).

Stark, M.

M. Stark, Application of Optical Fourier Transform (Academic, New York, 1982), Chap. 1.

Appl. Opt. (3)

J. Opt. Soc. Am (1)

E. B. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
[Crossref]

J. Opt. Soc. Am. (2)

Optik (1)

P. S. Naidu, D. V. B. Rao, Optik 55, 351 (1979).

Other (5)

M. Stark, Application of Optical Fourier Transform (Academic, New York, 1982), Chap. 1.

According to J. W. Goodman, use of the parallel beam system as an optical Fourier transform was reported first in E. Abbe, Arch. Mikrosk. Anat. Entwicklungsmech. 9, 413 (1873); A. B. Porter, Philos. Mag. (6) 11, 154 (1906). However, the first clear and systematic description of optical Fourier transform is probably P. M Duffieux, “L’intégrale de Fourier et ses applications à l’Optique”, Besançon (1946). All these authors observed the FT in the back focal plane of a lens thus working in parallel beam illumination. The question of the generalization of this setup to any type of source–Fourier plane conjugation is not clear; we found a use for the converging beam setup in A. Marechal and P. Croce, C. R. Acad. Sci. 237, 607 (1953). The question of optical performance was never examined, and the use of optical FT was restricted to very low spatial frequencies and low aperture.

K. Preston, in Optical and electro-optical information processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965), p. 68.

J. W. Goodman, in Optical Information Processing, Y. E. Nesterikhin, G. W. Stroke, W. Kock, Eds. (Plenum, New York, 1976), pp. 85–103.
[Crossref]

Such a corrected doublet is described in S. Lowenthal and Y. Belvaux, Rev. Opt. Theor. Instrum. 46, 1 (1967).

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Figures (5)

Fig. 1
Fig. 1

Two basic optical Fourier transform setups: (a) the f-f setup, (b) the converging-beam setup.

Fig. 2
Fig. 2

Geometry of the single spatial frequency hologram reconstruction. R is defined by R = αd. Grating and Fourier plane recall the correspondence with the initial Fourier transform problem.

Fig. 3
Fig. 3

Maximum wave front deviation in the grating plane and corresponding diffraction spot: Ast, astigmatism; FC, field curvature; C, coma. Setup parameters are: solid line, half-aperture α = 1/60; object radius R = 10 mm. This yields the left diffraction spot at 100 mm−1. Dotted line, α = 1/30; R = 20 mm. This yields the right diffraction spot at 100 mm−1; see text for a definition of the diffraction spot. With the same definition, the stigmatic spot would be a disk 21 μm in diam (left spot) and 10.4 μm in diam (right spot).

Fig. 4
Fig. 4

Diffraction-limited working range of the CB-FT. The aberration level is measured by the standard deviation σ (object size, aperture, frequency) of the wave front, compared to the best sphere. Curves are defined by σ = λ/10 for 100, 50, and 20 mm−1. Points under a particular curve define a setup which is diffraction-limited up to the corresponding spatial frequency.

Fig. 5
Fig. 5

Same aberration curves as in Fig. 3 when astigmatism and field curvature are corrected for one particular frequency, here νx = 100 mm−1. Parameters are α = 1/30, R = 20 mm. The diffraction spots are for νx = 50 mm−1 and νx = 100 mm−1. Note the enlarged scale of the diffraction spot with respect to Fig. 3. Here the stigmatic spot would be 10.4 μm in diam.

Tables (4)

Tables Icon

Table I Constraints for the Optical Component in the f-f and Converging-Beam Fourier Transforms

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Table II Aberrations in the FT Using the f-f Setup with a General-Purpose Complex Lens and the CB Setupa

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Table III Optical Specifications for Two Fourier Lenses (Tropel 700-6328 and 710-6328).

Tables Icon

Table IV Aberrations in the FT Using the f-f Setup with a Fourier Lens (710-6328 of Table III) and the CB Setupa

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